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I thank Thomas Richardson and James Robins for their discussion of my article, and discuss the similarities and differences between their approach to causal modelling, based on single world intervention graphs, and my own decision-theoretic approach.

The evidence-based medicine approach to causal medical inference is the dominant account among medical methodologists. Competing approaches originating in the philosophy of medicine seek to challenge this account. In order to see how successful these challenges are, we need to assess the performance of all approaches in real world medical inference. One important real world problem all approaches could be applied to is the assessment of drugs for approval by drug regulation agencies. This study assesses the success of the status quo against an empirical non-systematically obtained body of evidence and we scrutinise the alternative approaches from the armchair, contemplating how they would fare in the real world. We tentatively conclude that the status quo is regularly not successful at its primary task as it regularly fails to correctly assess effectiveness and safety and suggest that this is due to inherent factors of the “messy real world.” However, while all alternatives hold promise, they are at least as susceptible to the real world issues that beset the status quo. We also make recommendations for changes to current drug approval procedures, identify lacunae to fill in the alternatives, and finally, call for a continuation of the development of alternative approaches to causal medical inference and recommendations for changes to current drug approval procedures.

Personalized decision making targets the behavior of a specific individual, while population-based decision making concerns a subpopulation resembling that individual. This article clarifies the distinction between the two and explains why the former leads to more informed decisions. We further show that by combining experimental and observational studies, we can obtain valuable information about individual behavior and, consequently, improve decisions over those obtained from experimental studies alone. In particular, we show examples where such a combination discriminates between individuals who can benefit from a treatment and those who cannot – information that would not be revealed by experimental studies alone. We outline areas where this method could be of benefit to both policy makers and individuals involved.

I thank Judea Pearl for his discussion of my paper and respond to the points he raises. In particular, his attachment to unaugmented directed acyclic graphs has led to a misapprehension of my own proposals. I also discuss the possibilities for developing a non-manipulative understanding of causality.

In a recent issue of this journal, Philip Dawid (2021) proposes a framework for causal inference that is based on statistical decision theory and that is, in many aspects, compatible with the familiar framework of causal graphs (e.g., Directed Acyclic Graphs (DAGs)). This editorial compares the methodological features of the two frameworks as well as their epistemological basis.

I thank Ilya Shpitser for his comments on my article, and discuss the use of models with restricted interventions.

The multivariate skew elliptic distributions include the multivariate skew-t distribution, which is represented as a mean- and scale-mixture distribution and is useful for analyzing skewed data with heavy tails. In the estimation of location parameters in the multivariate skew elliptic distributions, we derive minimax shrinkage estimators improving on the minimum risk location equivariant estimator relative to the quadratic loss function. Especially in the skew-t distribution, we suggest specific improved estimators where the conditions for their minimaxity do not depend on the degrees of freedom. We also study the case of a general elliptically symmetrical distribution when the covariance matrix is known up to an unknown multiple, but a residual vector is available to estimate the scale. [ABSTRACT FROM AUTHOR]

This paper presents an axiomatic approach to finite Markov decision processes where the discount rate is zero. One of the principal difficulties in the no discounting case is that, even if attention is restricted to stationary policies, a strong overtaking optimal policy need not exists. We provide preference foundations for two criteria that do admit optimal policies: 0-discount optimality and average overtaking optimality. As a corollary of our results, we obtain conditions on a decision maker's preferences which ensure that an optimal policy exists. These results have implications for disciplines where dynamic programming problems arise, including automatic control, dynamic games, and economic development. [ABSTRACT FROM AUTHOR]

We give a new example for a proper scoring rule motivated by the form of Anderson–Darling distance of distribution functions and an example of Brehmer and Gneiting. [ABSTRACT FROM AUTHOR]

We develop a mathematical and interpretative foundation for the enterprise of decision-theoretic (DT) statistical causality, which is a straightforward way of representing and addressing causal questions. DT reframes causal inference as “assisted decision-making” and aims to understand when, and how, I can make use of external data, typically observational, to help me solve a decision problem by taking advantage of assumed relationships between the data and my problem. The relationships embodied in any representation of a causal problem require deeper justification, which is necessarily context-dependent. Here we clarify the considerations needed to support applications of the DT methodology. Exchangeability considerations are used to structure the required relationships, and a distinction drawn between intention to treat and intervention to treat forms the basis for the enabling condition of “ignorability.” We also show how the DT perspective unifies and sheds light on other popular formalisations of statistical causality, including potential responses and directed acyclic graphs.

A binary Wald sequential probability ratio test that uses the residuals of two norm-inequality-constrained Kalman filters for its likelihood ratio is employed for a class of compound hypothesis tests on non-stationary systems. The hypotheses concern an inequality constraint on the norm of some elements of the system state. Each of the two constrained Kalman filters minimizes the summed squares of its estimation errors subject to one or the other direction of the inequality constraint. This test is applied to the problem of spacecraft conjunction assessment, wherein the constraint concerns the close approach distance between a spacecraft and another space object. The outcome of the test can inform decisions concerning risk mitigation maneuvers by an active spacecraft. [ABSTRACT FROM AUTHOR]

Estimation of the scale parameter of the scale mixture of a location–scale family under the scale-invariant loss function is considered. The technique of Strawderman (Ann Stat 2(1):190–198, 1974) is used to obtain a class of estimators improving upon the best affine equivariant estimator of the scale parameter under certain conditions. Further, integral expressions of risk difference approach of Kubokawa (Ann Stat 22(1):290–299, 1994) is used to derive similar improvements for the reciprocal of the scale parameter. Using the improved estimator of the scale parameter and the improved estimator of the reciprocal of the scale parameter, classes of improved estimators for the ratio of scale parameters of two populations have been derived. In particular, Stein type and Brewster–Zidek type estimators are provided for the ratio of scale parameters of two mixture models. These results are applied to the scale mixture of exponential distributions, this includes the multivariate Lomax and the modified Lomax distributions. [ABSTRACT FROM AUTHOR]

This paper contributes to answering a question that is of crucial importance in risk management and extreme value theory: How to select the threshold above which one assumes that the tail of a distribution follows a generalized Pareto distribution. This question has gained increasing attention, particularly in finance institutions, as the recent regulative norms require the assessment of risk at high quantiles. Recent methods answer this question by multiple uses of the standard goodness-of-fit tests. These tests are based on a particular choice of symmetric weighting of the mean square error between the empirical and the fitted tail distributions. Assuming an asymmetric weighting, which rates high quantiles more than small ones, we propose new goodness-of-fit tests and automated threshold selection procedures. We consider a parameterized family of asymmetric weight functions and calculate the corresponding mean square error as a loss function. Then we explicitly determine the risk function as the expected value of the loss function for finite sample. Finally, the risk function can be used to discuss whether a symmetric or asymmetric weight function should be chosen. With this the goodness-of-fit test which should be used in a new method for determining the threshold value is specified. [ABSTRACT FROM AUTHOR]

In many real life situations, prior information about the parameters is available, such as the ordering of the parameters. Incorporating this prior information about the order restrictions on parameters leads to more efficient estimators. In the present communication, we investigate estimation of the ordered scale parameters of two shifted exponential distributions with unknown location parameters under a class of bowl-shaped loss functions. We have proved that the best affine equivariant estimator (BAEE) is inadmissible. Various non smooth and smooth estimators has been obtained which improve upon the BAEE. In particular we have derived the improved estimators for some well known loss functions. Finally numerical comparison is carried out to compare the risk performance of the proposed estimators. [ABSTRACT FROM AUTHOR]

The debate of which quantitative risk measure to choose in practice has mainly focused on the dichotomy between value at risk (VaR) and expected shortfall (ES). Range value at risk (RVaR) is a natural interpolation between VaR and ES, constituting a tradeoff between the sensitivity of ES and the robustness of VaR, turning it into a practically relevant risk measure on its own. Hence, there is a need to statistically assess, compare and rank the predictive performance of different RVaR models, tasks subsumed under the term "comparative backtesting" in finance. This is best done in terms of strictly consistent loss or scoring functions, i.e., functions which are minimized in expectation by the correct risk measure forecast. Much like ES, RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, we show that a triplet of RVaR with two VaR-components is elicitable. We characterize all strictly consistent scoring functions for this triplet. Additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares regression. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of estimating the quantile of a two-parameter exponential distribution with respect to an arbitrary strictly convex loss function under progressive type II censoring. Inadmissibility of the best affine equivariant (BAE) estimator is established through a conditional risk analysis. In particular we provide dominance results for quadratic, linex and absolute value loss functions. Further, a class of dominating estimators is derived using the IERD (integral expression of risk difference) approach of Kubokawa (1994). In sequel the generalized Bayes estimator is shown to improve the BAE estimator. [ABSTRACT FROM AUTHOR]

Men and jobs alike are characterized by a single trait, which may take on categorical values according to given population frequencies. Men arrive to the system following a Poisson process and wait till jobs are assigned to them. Jobs arrive to the system following another, independent, Poisson process. An arriving job must be assigned to a waiting man immediately, or be discarded, ensuing no gain. An assignment of a job to a man yields a higher gain if they match in trait, and a lower one if not. Each man waits a limited time for a job and leaves the system if unassigned by that time limit. It is stipulated that a man who arrives first has priority to either accept the pending job, or to pass it to the next man, who makes a similar decision. The last man in the line takes the job, or it is discarded. The individually optimal policy for each man is defined by some critical time for accepting a mismatched job. We solve for the critical times, depending on the mens' place in the queue, and obtain expressions for the ensuing optimal value functions of this system, for expected gain. The model originates from the utility-equity dilemma in assigning live organs to patients on the national waiting list. The paper reports numerical comparison of the above policy with alternative ones, for several performance measures. [ABSTRACT FROM AUTHOR]

This article presents two theorems: (1) a generalization of Löb's Theorem that applies to formal proof systems operating with bounded computational resources, such as formal verification software or theorem provers, and (2) a theorem on the robust cooperation of agents that employ proofs about one another's source code as unexploitable criteria for cooperation. The latter illustrates a capacity for outperforming classical Nash equilibria and correlated equilibria, attaining mutually cooperative program equilibrium in the Prisoner's Dilemma while remaining unexploitable, i.e., sometimes achieving the outcome (Cooperate, Cooperate), and never receiving the outcome (Cooperate, Defect) as player 1. [ABSTRACT FROM AUTHOR]

In this paper, assuming that the error terms follow a multivariate t distribution, we derive the exact formula for the predictive mean squared error (PMSE) of two different types of pretest estimators. It is shown analytically that one of the pretest estimator dominates the SR estimator if a critical value of the pretest is chosen appropriately. Also, we compare the PMSE of the pretest estimators with the MMSE, AMMSE, SR and PSR estimators by numerical evaluations. Our results show that the pretest estimators dominate the OLS estimator for all combinations when the degrees of freedom is not more than 5. Key Words: Predictive mean squared error; Homogeneous preliminary test estimator; Heterogeneous preliminary test estimator; Multivariate t error term. [ABSTRACT FROM AUTHOR]

We propose an extension of parametric product partition models. We name our proposal nonparametric product partition models because we associate a random measure instead of a parametric kernel to each set within a random partition. Our methodology does not impose any specific form on the marginal distribution of the observations, allowing us to detect shifts of behaviour even when dealing with heavy-tailed or skewed distributions. We propose a suitable loss function and find the partition of the data having minimum expected loss. We then apply our nonparametric procedure to multiple change-point analysis and compare it with PPMs and with other methodologies that have recently appeared in the literature. Also, in the context of missing data, we exploit the product partition structure in order to estimate the distribution function of each missing value, allowing us to detect change points using the loss function mentioned above. Finally, we present applications to financial as well as genetic data. [ABSTRACT FROM AUTHOR]

We analyze the problem of selecting the model that best describes a given dataset. We focus on the case where the best model is the one with the smallest error, respect to the reference data. To select the best model, we consider two components: (a) an error measure to compare individual data points, and (b) a function that combines the individual errors for all the points. We show that working with the most general definition of consistency, it is impossible to extend individual error measures in a way that provides a unanimous consensus about which is the best model. We also prove that, in the best case, modifying the notion of consistency leads to expressions that are too ill-behaved to be of any practical utility. These results show that selecting the model that best describes a dataset depends heavily on the way one measures the individual errors, even if these measures are consistent. [ABSTRACT FROM AUTHOR]

Improved point and interval estimation of the smallest scale parameter of n independent populations following two-parameter exponential distributions are studied. The model is formulated in such a way that allows for treating the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The classes of improved point and interval estimators are enriched with Stein-type, Brewster and Zidek-type, Maruyama-type and Strawderman-type improved estimators under both quadratic and entropy losses, whereas using as a criterion the coverage probability, with Stein-type, Brewster and Zidek-type, and Maruyama-type improved intervals. The sampling framework considered incorporates important life-testing schemes such as i.i.d. sampling, type-II censoring, progressive type-II censoring, adaptive progressive type-II censoring, and record values. [ABSTRACT FROM AUTHOR]

In estimating p(⩾ 2) independent Poisson means, Clevenson and Zidek (1975) have proposed a class of estimators that shrink the unbiased estimator to the origin and dominate the unbiased one under the normalized squared error loss. This class of estimators was subsequently enlarged in several directions. This article deals with the problem and proposes new classes of dominating estimators using prior information pertinently. Dominance is shown by partitioning the sample space into disjoint subsets and averaging the loss difference over each subset. Estimation of several Poisson mean vectors is also discussed. Further, simultaneous estimation of Poisson means under order restriction is treated and estimators which dominate the isotonic regression estimator are proposed for some types of order restrictions. [ABSTRACT FROM AUTHOR]

Shafer's belief functions were introduced in the seventies of the previous century as a mathematical tool in order to model epistemic probability. One of the reasons that they were not picked up by mainstream probability was the lack of a behavioral interpretation. In this paper, we provide such a behavioral interpretation and re-derive Shafer's belief functions via a betting interpretation reminiscent of the classical Dutch Book Theorem for probability distributions. We relate our betting interpretation of belief functions to the existing literature. [ABSTRACT FROM AUTHOR]

For an arbitrary strictly convex loss function, we study the problem of estimating a linear parametric function μ + kσ, k is a known constant, when a doubly censored sample is available from a two-parameter exponential E(μ, σ) population. We establish the inadmissibility of the best affine equivariant (BAE) estimator by deriving an improved estimator. We provide various implications for quadratic and linex loss functions in detail. Improvements are obtained for the absolute value loss function as well. Further a new class of estimators improving upon the BAE estimator is derived using the Kubokawa method. This class is shown to include some benchmark estimators from the literature. [ABSTRACT FROM AUTHOR]

Traditional phase III clinical trials are powered to detect an overall treatment effect. However, it has increasingly been shown that many treatments are effective only for a subset of a population. The adaptive signature design uses genomic/proteomic information to prospectively predict a subset of patients more sensitive to treatment. Tests for overall treatment effect and for treatment effect in the predicted subset are conducted. In this work properties of the adaptive signature design are investigated through simulation. It was found that models which excluded expression main effect terms had higher empirical power than models which included them. [ABSTRACT FROM AUTHOR]

Estimation of two normal means with an order restriction is considered when a covariance matrix is known. It is shown that restricted maximum likelihood estimator (MLE) stochastically dominates both estimators proposed by Hwang and Peddada [Confidence interval estimation subject to order restrictions. Ann Statist. 1994;22(1):67–93] and Peddada et al. [Estimation of order-restricted means from correlated data. Biometrika. 2005;92:703–715]. The estimators are also compared under the Pitman nearness criterion and it isshownthat the MLE is closer to ordered means than the other two estimators. Estimation of linear functions of ordered means is also considered and a necessary and sufficient condition on the coefficients is given for the MLE to dominate the other estimators in terms of mean squared error. [ABSTRACT FROM PUBLISHER]

This paper is concerned with additive models of the form Y = M + E , where Y is an observed n × m matrix with m < n , M is an unknown n × m matrix of interest with low rank, and E is a random noise whose distribution is elliptically symmetric. For general estimators M ̂ of M , we develop unbiased risk estimates, including in the special case where E is Gaussian with covariance matrix proportional to the identity matrix. To this end, we develop a new Stein–Haff type identity. We apply the theory to a model selection framework with estimators defined through a soft-thresholding function. We establish the robustness of our approach within a large subclass of elliptical distributions. [ABSTRACT FROM AUTHOR]

The usual methods of estimating the unknown parameters of a distribution, use only the information given from the sample data. In many cases, there is, also, another important information for estimating the unknown parameters of our model, such as the order of these parameters, and this last information improves the quality of estimation. In this paper, we deal with the problem of estimating the ordered scale parameters from two populations of the multivariate Lomax distribution, with unknown location parameters. It is proved that the best equivariant estimators of the scale parameters (in the unrestricted case) are not admissible and we construct estimators that improve upon the usual ones (when these parameters are known to be ordered). [ABSTRACT FROM AUTHOR]

Rating scales are used to elicit data about qualitative entities (e.g., research collaboration). This study presents an innovative method for reducing the number of rating scale items without the predictability loss. The 'area under the receiver operator curve method' (AUC ROC) is used. The presented method has reduced the number of rating scale items (variables) to 28.57% (from 21 to 6) making over 70% of collected data unnecessary. Results have been verified by two methods of analysis: Graded Response Model (GRM) and Confirmatory Factor Analysis (CFA). GRM revealed that the new method differentiates observations of high and middle scores. CFA proved that the reliability of the rating scale has not deteriorated by the scale item reduction. Both statistical analysis evidenced usefulness of the AUC ROC reduction method. [ABSTRACT FROM AUTHOR]

We consider estimation of the inverse scatter matrices Σ − 1 for high-dimensional elliptically symmetric distributions. In high-dimensional settings the sample covariance matrix S may be singular. Depending on the singularity of S , natural estimators of Σ − 1 are of the form a S − 1 or a S + where a is a positive constant and S − 1 and S + are, respectively, the inverse and the Moore–Penrose inverse of S . We propose a unified estimation approach for these two cases and provide improved estimators under the quadratic loss tr ( Σ ˆ − 1 − Σ − 1 ) 2 . To this end, a new and general Stein–Haff identity is derived for the high-dimensional elliptically symmetric distribution setting. [ABSTRACT FROM AUTHOR]

Based on Stein’s famous shrinkage estimation of a multivariate normal distribution, we propose a new type of estimators of the distribution function of a random variable in a nonparametric setup. The proposed estimators are then compared with the empirical distribution function, which is the best equivariant estimator under a well-known loss function. Our extensive simulation study shows that our proposed estimators can perform better for moderate to large sample sizes. [ABSTRACT FROM PUBLISHER]

This paper presents a result that provides a positive answer to the question of existence of regularities of the so called random in a broad sense mass phenomena (Kolmogorov, ). The theorem of existence of statistical regularities of mass phenomena in the form of closed in weak-* topology families of finitely-additive probability distributions, and their significance to decision theory, constitute the content of this paper. [ABSTRACT FROM AUTHOR]

Whereas various efficient learning algorithms have been recently proposed to perform bipartite ranking tasks, cast as receiver operating characteristic (ROC) curve optimisation, no method fully tailored toK-partite ranking whenK≥3 has been documented in the statistical learning literature yet. The goal is to optimise the ROC manifold, or summary criteria such as its volume, the gold standard for assessing performance inK-partite ranking. It is the main purpose of this paper to describe at length an efficient approach to recursive maximisation of the ROC surface, extending theTreeRankmethodology originally tailored for the bipartite situation (i.e. whenK=2). The main barrier arises from the fact that, in contrast to the bipartite case, the volume under the ROC surface criterion of any scoring rule takingK≥3 values cannot be interpreted as acost-sensitivemisclassification error and no method is readily available to perform the recursive optimisation stage. The learning algorithm we propose, calledTreeRank Tournament(referred to as ‘TRT’ in the tables), breaks it and builds recursively an ordered partition of the feature space. It defines a piecewise scoring function whose ROC manifold can be remarkably interpreted as a statistical version of an adaptive piecewise linear approximant of the optimal ROC manifold. Rate bounds in sup norm describing the generalisation ability of the scoring rule thus built are established and numerical results illustrating the performance of the TRT approach, compared to that of natural competitors such as aggregation methods, are also displayed. [ABSTRACT FROM AUTHOR]